Page 59 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

50 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

Substituting (3.52)and (3.53)into(3.51)results in

2 2 2 1 T 2 T 1 2
˜
˜
˜
˜
˙ V ≤−r 1k 1z − r 2k 2z − r 3k 3z − − 1 +
1 2 3 1
2 2 2
2
2 2
¯
+ + r iM δ i i (3.54)
1
2
i=1
≤−π V + ι
−1 −1
where π = min 2r 1mk 1 ,2r 2mk 3 ,2r 3mk 3 , 1 /λ max ( ), 2 /λ max ( ) , ι =

1
2
2
1 + 2 + 2 r ¯
2 2 1 i=1 iM δ i i are all positive constants with r im ,r iM > 0,i =
1,2,3 are the minimum and maximum values of r i ,i = 1,2,3.
From (3.54) and Lyapunov theorem, we know V(t) eventually con-
verges to a small set bounded by ι/π in which the ESN approximation is
valid. Therefore, all error signals (e.g., s i , 1 , ) are semi-globally uniformly
˜
˜
and ultimately bounded. The adaptive parameters ˆ ˆ , the system state
, 1
x i and control signals u, ¯χ i are all bounded. Specifically, we can verify that
√
|z i |≤ 2ι/π as t →∞. Consequently, based on the property of PPF func-
tion μ i (t) and the error transform function S i (s i /μ i ), we know that the
tracking error z i (t) will be strictly guaranteed within the bounds defined
by (3.13). This completes the proof.
3.4 SIMULATION AND EXPERIMENT
3.4.1 Simulation Results
In this section, we first provide numerical simulations to validate the pro-
posed control for the two-inertia system as shown in Fig. 3.1. The system
parameters in the model (3.1)are givenas J m = 0.005 kg m, J l = 0.04 kg m,
k f = 5as [34]. The LuGre model is used to simulate the friction dynam-
ics. The PPF parameters are chosen as μ i0 = 1.2, μ i∞ = 0.1, κ i = 0.5, and
δ =−1, δ i = 1.5. For the used ESNs, the number of neurons in the in-
i
put and hidden layers are set as 2 and 13, respectively. The initial weight
(0) is set as zero. Other parameters in the DSC with HGDT and ob-
ˆ
server are given as ρ 1,i = ρ 2,i = 1(i = 1,2,3), H = 100, α = 1/2, β = 2/3,
1 = 2 = 0.1, k 1 = 5,k 2 = 40,k 3 = 25.
In order to illustrate the effect of the parameter uncertainties and exter-
nal disturbance on the transient performance of the control system, a step
signal d = 1 is added to simulation at time t = 0 s. Simulation results of
the proposed control scheme are shown in Fig. 3.5, which shows that sat-
isfactory control performance can be guaranteed even in the presence of